"In most sciences, one generation tears down what others have built, and what one has established, another undoes. In mathematics alone each generation adds a new story to the old structure." -- (Hermann Hankel, 1839-1873) Johann Kepler (1571-1630): Despite a life that involved may misfortunes (he was accused of heresy, his child died of smallpox, his wife went mad and died, his mother was accused of witchcraft), the German-born Kepler made many contributions to mathematics and astronomy. Deeply religious, he angered the Church by supporting the Copernican theory that the earth was not the center of the universe. His greatest accomplishment was establishing three laws of planetary motion. He was the first to suggest that the paths of the planets are elliptical rather than circular. When Herkimer was a lawyer, where did he suggest a mermaid take a grievance? Answer: To Small Clams Court. Things Herky would like to know: If your conscious is always clear, is that the sign of a bad memory? If Barbie is so popular, why do you have to buy her friends? ASSIGNMENT #35 Reading: Section 6.2 (pages 317-322). Exercises: Test 5B (Handout in class).

You are in Section 6.2 and working on problems(Test 5B) from Chapter 5.

Key words and phrases in the reading:

 sample space tree diagram sampling with replacement sampling without replacement

Thinking back to Chapter 5, the conceptsof stratification and blocking are often confused.

Stratificationexample: Cate Headmaster Benjamin Williams wants tochoose a sample of eight Cate students to represent the School at aconference. Since he wants each class to be represented, hestratifies the student population by class, and then randomly choosestwo students from each strata (seniors, juniors, sophomores,freshmen). This is a stratified randomsample, but note that it isnot asimple random sample (SRS) of eight Cate students. In this hypothetical example, Mr. Williams would bestratifying to achieve representation from each class. The selectionprocess would not allow for the committee to be void of juniors, forinstance, whereas a SRS could produce a set of eight students void ofjuniors.

Blocking example: Mr. Ned Bowler wants to see if adding extra nitrogen to soil would increase the yield for a specific vegetable. He takes his rectangular plot and divides it into sixteen blocks, as shown in the diagram at the right. He then forms eight strata (described below).

 STRATA # Blocks 1 1 and 2 2 3 and 4 3 5 and 6 4 7 and 8 5 9 and 10 6 11 and 12 7 13 and 14 8 15 and 16

Now, Mr. Bowler uses a random process to choose one block from each strata, obtaining a set of eight blocks that we will call S1. The set of remaining eight blocks will be called S2. He will now flip a coin, and apply nitrogen to the blocks in S1 if he gets a head; otherwise the nitrogen will be applied to the blocks in S2. At a later time, he will check the crop yield results from S1 and S2 to see if they are significantly different.

Blocking is done to reduce variation. AP Statistics students should realize that the blocking design described is statistically superior to a design that simply divides the field into two plots and applying nitrogen to one of the plots.

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Text:
The Practice of Statistics, by Yates, Moore, McCabe. New York,W.H. Freeman and Company, 1999. (ISBN 0-7167-3370-6)

Supplemental books:
The Cartoon Guide to Statistics, by Gonick and Smith. NewYork, HarperCollins Publishers, 1993. (ISBN 0-06-273102-5)
How to Lie with Statistics, by Darrell Huff. New York, W.W.Norton & Company, 1982 (ISBN 0-393-09426-X)